Abstract
This paper is intended as a consideration of metaphorical understanding of fraction in terms of both its form and content. As for the representation of fraction in European style, it is composed of cardinal number and ordinal number like as m-nths. They are extremely different each other in the epistemological genesis. Ordinarity is the index of discrimination. On the other hand, cardinarity is the product of categorizing same. The combination of such an opposite concept reflects the rhetorical intention rather than supports the logical understanding. The fraction in Europe was not originally invented as the object for mathematics but as the means for trade and commerce in the ancient Greece. It, therefore, asked for the metaphorical understanding from the beginning. For instance, "two" means "2/3", and "third" means "1/3" in the line of Iriad K,253 by Homer: "Two parts of the night are past, the third part remains." (Van der Waerden, 1954, p. 50) The former trope by cardinal number could be considered as metaphor because of similarity or analogy relation between two numbers. The latter trope by ordinal number could be regarded as metonymy because the part represents the whole. The point is that the fraction is in itself figurative form, therefore is metaphorical concept. As for the meaning of fraction, there is difference between the literal meaning and the mathematical meaning. It causes metaphorical understanding, which helps to get the sound comprehension of the mathematical concept. This is the conclusion of the didactical analysis based on the Transcendental Recursive Model of Understanding Mathematics by Pirie and Kieren (1994a). They insist that Image Having is characterized by the speech of metaphor and Property Noticing by the speech of simile. Mathematical understanding, therefore, proceeds to simile from metaphor. This seems to be the correct insistence on the development of metaphorical understanding. Their assertion, however, could be criticized by the complementarity in mathematical understanding implied by Dorfler (1984, p. 261) as follows: P1. The transcendent recursive model does not give us any explanation about the system of modification in cognitive paradigm from metaphor to simile. It just corresponds metaphor with image having level and simile with property noticing level. P2. The consideration on understanding mathematics by Pirie and Kieren is inclined to the analysis of relation i.e. die Beziehungsschem and as a result leaves the analysis of activity i.e. die Handlungsschemas in terms of Darner. In the learning process of subjects Don and Sam in their paper, they just focus on the former. The author has introduced the viewpoint of metonymy based on contiguity to answer those problems P1 and P2. This introduction is justified by the above complementarity. A1. The intellectual renovation from image having to property noticing could not be clarified without the consideration on Handlungsschema. The roll of it is analogous to the cognitive roll of metonymy and synecdoche according to Zawatowski (1985, pp.262-263). Activity of children, therefore, should be analysed from their viewpoints. A2. The activity of subjects, Don and Sam, in the paper of Pirie and Kieren could be analysed clearly by metonymy. Their cognitive level should be consequently similical rather than metaphorical although the paper say that it is metaphorical.
